What was the first sentient race that awoke on Arda?

Question

Some background: Tolkien's history of elves singing and ents walking and talking.

One of the foundations of Tolkien's mythology is the Elves being the Firstborn. The Dwarves are even put back to sleep until the Elves awake. I've always taken this to mean the first born sentient race¹.

However, several quotes are making me question this:

Even some whole inventions like the remarkable Ents, oldest of living rational creatures, Shepherds of the Trees, are omitted.

^{The Letters of J.R.R. Tolkien, To Milton Waldman (#131), p. 160}

Ents. The most ancient people surviving in the Third Age were the Onodrim or Enyd. Ent was the form of their name in the language of Rohan.

^{The Lord of the Rings: Return of the King, Appendix F: The Languages and Peoples of the Third Age, p. 511}

[...]before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West."

^{The Silmarillion, Of Aulë and Yavanna, p. 41}

There's another quote from the Silmarillion abouts Ents that contradicts the ones above:

When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar[...]

^ibid.

Concerning the Dwarves, Ilúvatar says this:

But I will not suffer this: that these should come before the Firstborn of my design, nor that thy impatience should be rewarded. They shall sleep now in the darkness under stone, and shall not come forth until the Firstborn have awakened upon Earth

^{The Silmarillion, Of Aulë and Yavanna, p. 38}

So, are Elves not the first sentient beings to awaken? Were they only called "Firstborn" in relation to Men? Was it the Ents? Or maybe Eagles?

I am obviously excluding from this any race that was created outside Arda (Ainur, Maiar) or weird individuals (Bombadil, Ungoliant).

^{1"race", "species", "ethnicity" or whatever. Not trying to start a semantics war.}

^{All citations from the Harper Collins editions}

Answer 1

Ok, so here's what happened, in order:

1. Eru Ilúvatar would create Elves and then Men in the coming of the Children of Ilúvatar. Aulë, not wanting to wait until the coming of the children to have beings to whom he could teach his crafts, created 7 Dwarves in secret. But Aulë did not actually have the power to create life, so these Dwarves were not technically a sentient species.
2. Ilúvatar finds out about these Dwarves and reprimands Aulë for his impatience. Aulë realises he has made a mistake and offers up the Dwarves for Ilúvatar to do with as he will, including destroy them if he so wished.
3. Ilúvatar accepts Aulë's offering and grants the 7 Dwarves life, so when Aulë attempts to destroy his creations, they cower in fear and beg for mercy, which Aulë grants them.
4. Ilúvatar decrees that he shall not suffer Dwarves to exist before his children, and so the 7 Dwarves shall sleep underground until his children awaken and walk Middle-earth.
5. Yavanna voices concern for her creations, the plants and animals, for she fears the Dwarves would fell them to fuel their industrious nature. Aulë, in defense of his Dwarves, informs her that Ilúvatar's children would also have need of trees. She then asks Manwë for counsel and asks that her creations be protected. When asked which she would have protection granted to, she replies that all things have their worth, but that trees should punish those who wrong them.
6. Manwë informs Ilúvatar of Yavanna's request and Ilúvatar grants it. It is said that when the Firstborn (The Elves) awaken, spirits shall descend to Middle-earth and inhabit beings within it, guardians who will protect the plants and animals. This is the conception of both the Ents and the Great Eagles.

So in fact many races were created long before the Elves ever existed, but all were asleep before the Elves awoken. The first race to awaken on Middle-earth were the Elves. Technically speaking, the very first race to "awaken" on Arda were the Dwarves, as they were created first by Aulë and granted life by Ilúvatar before they were put to sleep.

EDIT: Extra bit to answer OP's last question at the bottom of their post. Yes, the term "Firstborn" is applied to Elves only in relation to men. Men are, in comparison, known as the "Afterborn".

Answer 2

There is a distinction between species in Middle-earth that provides the answer here, and that is the distinction between what is called "races" in the question, and Children of Ilúvatar; these are not the same thing.

The Ainulindalë provides a definition for Children of Ilúvatar:

For the Children of Ilúvatar were conceived by him alone; and they came with the third theme, and were not in the theme which Ilúvatar propounded at the beginning, and none of the Ainur had part in their making.

Elves are therefore the Firstborn Children of Ilúvatar which is confirmed by the Index to The Silmarillion:

Firstborn, The: The Elder Children of Ilúvatar, the Elves.

But none of this precludes the existence of other species who are not Children of Ilúvatar, whether they be sentient or not; i.e. species that both Ilúvatar and the Ainur had a hand in the making of.

Answer 3

Is possibly a word play?

None of the other examples that you mention, Ents or eagles, give birth. Ents may germinate, eagles hatch from eggs. The initial Dwarves were created and were asleep. They did not reproduce until after they were re awoken

Elves do give birth and so were possibly the 'first born' in a literal sense.